Drawing Twelfth Dimensional Objects Made Easy

September 12, 2008 · Print This Article

Drawing Twelfth Dimensional Objects Made Easy

Lately I have had a lot of students asking about what exponential values really mean. Being that exponents are one of the five components to real numbers I thought it necessary to explain what these mean and how we might understand them for dimensional values higher than three – especially since they will be all over the SAT Math, ACT Math, and SAT II Math tests.

Drawing twelfth dimensional objects is possible by expanding our understanding of the concept of dimensions themselves. As we add dimensions to units of measurement, what we are doing is, simply, creating an expansion of space. As part of Stephen Hawking’s analysis of Rene Descartes’ ingenuity, (from the book God Created The Integers) he tells us:

“We can easily see the power of Descartes’ mathematical methods with his exposition of two very basic mathematical operations: determining products and square roots.”

(?!) Really?!?!?! Descartes was able to accomplish so much by expanding the concepts of simple multiplication and exponential functions? That’s like building a house with a hammer and a screwdriver!!!

Let’s look a bit further into what he says next [bold and caps for emphasis]:

“Recall that to Euclid and his Greek successors all the way up through the sixteenth century, arithmetic propositions are stated in terms of geometrical figures such as line lengths, not because line lengths provide a good method for representing numbers, but because THAT IS WHAT NUMBERS ARE!”

WHOA! Stephen Hawking really did just say that!!! Numbers are representative of value and amount – this is true. But what numbers really are relates to physical, visual manifestations of value in the form of geometrical objects. As an example, numbers simply represent 1-Dimensional line lengths:

He explains further:

“Thus, determining the product (multiplying two numbers together) of two abstract values X and Y meant constructing a rectangle whose sides had length X and Y and realizing their product in the area of the rectangle!

As an example, if I wanted to perform 2*3*4*5, I can represent this idea in the form of a cubic structure that looks like TWO blocks stacked together with each block having dimensions THREE by FOUR by FIVE:

“Similarly, the ancients would take the square root of an abstract value X by representing X as a two dimensional figure and finding a square of equal area. The side of that square realized the square root of X.”

In other words, if we wanted to solve for a square root, cube root, or any other root – how can we accomplish this? We do so by expanding the 1 dimensional distance into the second (figure 1) and third dimensions (figure 2) (ie, setting the AREA OF A SQUARE (two dimensional object) equal to the length of a line, setting the VOLUME OF A CUBE (three dimensional object) equal to the length of a line, etc.). The side length of the smallest unit contained within the shape is the root.

There are misconceptions with multiple dimensions, and this probably comes from a misunderstanding of what the word “dimension” means. Descartes, himself, admits to his confusion:

“We should also note that those proportions that form a continuing sequence are to be understood in terms of a number of relations; others try to express these proportions in ordinary algebraic terms by means of several different dimensions and shapes. The first they call the root, the second the square, the third the cube, the fourth the biquadratic, and so on. These expressions have, I confess, long misled me. … All such names should be abandoned as they are liable to cause confusion in our thinking. For though a magnitude may be termed a cube or a biquadratic, it should never be represented to the imagination otherwise than as a line or a surface. … What, above all, requires to be noted is that the root, the square, the cube, etc, are merely magnitudes in continued proportion …”

By dimension in this sense, he tells us that it is to mean, “An order of magnitude.” So, as we add additional dimensions greater than 3, what we are essentially doing is compounding our cubic shapes into higher orders of magnitude within space. As an example, multiplying a number by itself three times creates a cube (remember, we take a 1 dimensional length and set it equal to the three dimensional volume of a cube. The length of the side of this cube is the CUBE ROOT of the line length’s value):

By Multiplying the number by itself the fourth time we are extending that cube in one direction to form a length of cubes:

By adding a 5th and 6th dimension I don’t just add a LENGTH of cubes, I add a WIDTH and HEIGHT of cubes also to form another cube!
Notice in the drawing above that we extend the cube out to a “length of three cubes” since our root value in the first place is 3. If this were 4^4 we would extend FOUR cubes (of length, width, and height equal to FOUR) into a length.

This actually makes a lot of sense as we look at, algebraically, what happens:

[a^6 = a^3 * a^3 = cube * cube = cube OF a cube]

Notice that “a” represents the root value (that is, the side length) and the exponent tells you how many orders of magnitude (dimension) to extend the root value.

So as we expand our thinking to higher dimensional representations of numbers, let us not forget that roots represent the collapse of these higher order of magnitudes BACK INTO A LINE LENGTH. These higher dimensional objects look like nested cubic structures as shown below. How many cubes will there be along the length, width, and height of these larger, nested cubes? The answer to this is simple: Whatever the root value is.

As an example, if we were to draw out what 8^6 looks like, it would not be a set of cubes of side length 8 that are stacked three by three by three since that would, algebraically, look like 8*8*8*3*3*3. In fact, the cubes would be stacked 8 long, wide, and high. Nevertheless, it will be a series of nested cubes and will look similar to the drawing below:

Thus, we are able to envision what a twelfth dimensional object looks like as well as any other dimensional object.

The great thing about math is that it is a logical subject. We can define our language very specifically and build upon those ideas so that we may fully understand seemingly difficult concepts the easy way. Understanding multiplication and exponents this way should give any student confidence on the SAT Math section, the ACT Math section, or the SAT II Math subject tests.

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